Question

Magnetic flux linked with a stationary loop of resistance `R` varies with respect to time during the time period `T` as follows:
`phi=aT(T-r)`
Find the amount of heat generated in the loop during that time. Inductance of the coil is negligible.
A. `(aT)/(3R)`
B. `(a^(2)T^(2))/(3R)`
C. `(a^(2)T^(2))`
D. `(a^(2)T^92))/(3R)`

Answer 1

Correct Answer - D
Given that `phi=at(T-t)` Induced e.m.f., `E=(dphi)/(dt)=(d)/(dt)[at(T-t)]`
`=at(0-1)+a(T-t)`
`=a(T-2t)`
So, induced emf is also a function of time.
:. Heat genrated in time `T` is
`H=int_(0)^(T)(E^(2))/(R )dt=(a^(2))/(R )int_(0)^(T)(T-at)^(2)dt`
`=(a^(2))/(R )int_(0)^(T)(E^(2))/(R )dt=(a^(2))/(R )int_(0)^(T)(T-at)^(2)dt`
`=(a^(2))/(R )int_(0)^(T)(T^(2)+4t^(2)-4tT)dt=(a^(2)T(3))/(3R)`